572 Найдите: а) һ, а и b, если b = 25, а = 16; 6) h, а и ь, если b=36, а = 64; в) а, с и ас, если b = 12, b = 6; г) 6, си бе, если а=8, а = 4; д) һ, b, ас и вс, если а = 6, c=9.

Решение:

В прямоугольном треугольнике ABC с прямым углом C и высотой CH, опущенной на гипотенузу, имеют место следующие соотношения:

• a² = ac ⋅ c
• b² = bc ⋅ c
• h² = ac ⋅ bc
• a ⋅ b = c ⋅ h

а) Дано: b = 25, ac = 16.

• b² = bc ⋅ c → 25² = bc ⋅ c
• ac + bc = c → 16 + bc = c
• Подставим c = 16 + bc в первое уравнение: 625 = bc ⋅ (16 + bc)
• bc² + 16bc - 625 = 0
• Решим квадратное уравнение относительно bc: D = 16² - 4 ⋅ 1 ⋅ (-625) = 256 + 2500 = 2756
• bc = (-16 ± √2756) / 2 = (-16 ± 2√689) / 2 = -8 ± √689. Так как bc > 0, то bc = √689 - 8 ≈ 18.27
• c = ac + bc = 16 + √689 - 8 = 8 + √689 ≈ 34.27
• a² = ac ⋅ c = 16 ⋅ (8 + √689) = 128 + 16√689 ≈ 548.32
• a = √(128 + 16√689) ≈ 23.42
• h = a ⋅ b / c ≈ (23.42 ⋅ 25) / 34.27 ≈ 17.04

б) Дано: bc = 36, ac = 64.

• c = ac + bc = 64 + 36 = 100
• a² = ac ⋅ c = 64 ⋅ 100 = 6400 → a = √6400 = 80
• b² = bc ⋅ c = 36 ⋅ 100 = 3600 → b = √3600 = 60
• h = a ⋅ b / c = 80 ⋅ 60 / 100 = 48

в) Дано: b = 12, bc = 6.

• b² = bc ⋅ c → 12² = 6 ⋅ c → c = 144 / 6 = 24
• a² = c² - b² = 24² - 12² = 576 - 144 = 432 → a = √432 = 12√3 ≈ 20.78
• ac = c - bc = 24 - 6 = 18

г) Дано: a = 8, ac = 4.

• a² = ac ⋅ c → 8² = 4 ⋅ c → c = 64 / 4 = 16
• b² = c² - a² = 16² - 8² = 256 - 64 = 192 → b = √192 = 8√3 ≈ 13.86
• bc = c - ac = 16 - 4 = 12

д) Дано: a = 6, c = 9.

• b² = c² - a² = 9² - 6² = 81 - 36 = 45 → b = √45 = 3√5 ≈ 6.71
• a² = ac ⋅ c → 6² = ac ⋅ 9 → ac = 36 / 9 = 4
• b² = bc ⋅ c → (3√5)² = bc ⋅ 9 → bc = 45 / 9 = 5
• h = a ⋅ b / c = 6 ⋅ 3√5 / 9 = 2√5 ≈ 4.47

Ответ:

• а) h ≈ 17.04, a ≈ 23.42, b = 25
• б) h = 48, a = 80, b = 60
• в) a ≈ 20.78, c = 24, ac = 18
• г) b ≈ 13.86, c = 16, bc = 12
• д) h ≈ 4.47, b ≈ 6.71, ac = 4, bc = 5